On approximative embeddability of diffeomorphisms in C¹-flows

Abstract

A function $f:I\to I$ is said to be C¹-embeddable if there exists a C¹-flow (iteration group) $\{f^t:I\to I,t\in R\}$ such that $f^1=f$. The C¹-embeddability on a compact interval I is a rare property. It is known that even $C^{\infty }$-diffeomorphisms with two hyperbolic fixed points need not be C¹-embeddable. However, every $C^r$-diffeomorphism, for $r\ge 2$, with one hyperbolic fixed point is uniquely embeddable in a $C^r$-flow. We consider the problem how to correct a given diffeomorphism with two hyperbolic fixed points making it C¹-embeddable. We prove that if $f\in D\text{iff}{}^2\left[0,1\right]$, $0<f\left(x\right)<x$ in $\left(0,1\right)$ and 0 and 1 are hyperbolic fixed points, then for every $a\in \left(0,1\right)$ and $ϵ>0$ and every diffeomorphism g such that $\text{supp} \left(f,-,g\right)\subset \left(a,-,ϵ,,,a,+,ϵ\right)$, $g\left(a\right)=f\left(a\right)$ and $g\prime \left(a\right)=f\prime \left(a\right){\theta }_f$ for a suitable chosen ${\theta }_f,$ there exists a unique C¹-embeddable function $\tilde{f}$ such that $f=\tilde{f}$ in $\left[0,1\right]\setminus \left(a,-,ϵ,,,f^{-1},\left(a\right)\right)$ and $\tilde{f}=g$ in $\left[a,-,ϵ,,,a\right]$. We determine the coefficient ${\theta }_f$ and we give a necessary and sufficient condition for the best C¹-embeddable approximation of f that is such that g = f.

Keywords::

• iteration groups
• flows
• dynamical systems
• functional equations

Keywords::

• 39B12
• 26A18
• 37C15